192 Chapter 6. Cold plasma waves in a magnetized plasma
- Bounded volume with no resonance andn^2 > 0 at some point in the bounded volume.
Sincen^2 =0occurs only at the bounding surfaces andn^2 →∞only at resonances,
n^2 must be positive and finite at everyθfor each location in the bounded volume.
The wave normal surface is thus ellipsoidal with symmetry about bothθ= 0and
θ=π/ 2 .The ellipse may deform as one moves inside the bounded volume, but will
always have the morphology of an ellipse. This type of wave normal surface is shown
in Fig.6.3(a). The wave normal surface is three dimensional and is azimuthally sym-
metric about thezaxis. - Bounded volume having a resonance at some angleθreswhere 0 <θres<π/ 2 and
n^2 (θ)positive forθ<θres.Atθres,n→∞so the radius of the wave normal surface
goes to zero. Forθ < θres, the wave normal surface exists (nis pure real since
n^2 > 0 ) and is plotted. At resonancesn^2 (θ)passes from−∞to+∞or vice versa.
This type of wave normal surface is a dumbbell type as shown in Fig. 6.3(b). - Bounded volume having a resonance at some angleθreswhere 0 <θres<π/ 2 and
n^2 (θ)positive forθ>θres.This is similar to case 2 above, except that now the wave
normal surface exists only for angles greater thanθresresulting in the wheel type
surface shown in Fig.6.3(c).
Consider now the relationship between the two modes (plus and minus sign) given by
Eq.(6.49). Because the two modes cannot intersect (cf. theorem 5) at angles other than
θ= 0,π/ 2 and mirror angles, if one mode is an ellipsoid and the other has a resonance
(i.e., is a dumbbell or wheel), the ellipsoid must be outside the other dumbbell or wheel;
for if not, the two modes would intersect at an angle other thanθ=0orθ=π/ 2 ). This is
shown in Figs. 6.3(d) and (e).
Also, only one of the modes can have a resonance, so at most one mode in a bounded
volume can be a dumbbell or wheel. This can be seen by noting that a resonance occurs
whenA→ 0 .In this caseB^2 >>| 4 AC|in Eq.(6.49) and the two roots arewell-separated.
This means that the binomial expansion can be used on the square root in Eq.(6.49)to obtain
n^2 ≃B±(B− 2 AC/B)
2 A
n^2 + ≃B
A
, n^2 −≃C
B
(6.55)
where|n^2 +|>>|n^2 −|sinceB^2 >>| 4 AC|.The rootn^2 + has the resonance and the root
n^2 −has no resonance. Since the wave normal surface of the minus root has no resonance,
its wave normal surface must be ellipsoidal (if it exists). Because the ellipsoidal surface
must always lie outside the wheel or dumbbell surface, the ellipsoidal surface will have a
larger value ofω/kcthan the dumbbell or wheel at everyθand so the ellipsoidal mode
will always be thefastmode. The mode with the resonance will be a dumbbell or wheel,
will lie inside the ellipsoidal surface, and so will always be theslowmode. This concept
of well-separated roots is quite useful and, if the roots are well-separated, then Eq.(6.47)
can be solved approximately for the large root (slow mode) by balancing the first two terms
with each other, and for the small root (fast mode) by balancing the last two terms with
each other.
Parameter space is subdivided into thirteen bounded volumes, each potentially con-